Helping in the kitchen at home through the years has resulted in the applicants' noticing two interesting phenomena. First, when a flat-bottom pot ages, its bottom often tends to bulge upward. Second, when the pot is heated from below by open flames, there is a tremendous amount or heat loss around the side of the pot.
The bulging phenomenon is somewhat contrary to common sense. As in the case of bi-metals used in thermostats and watch pendulums, one would expect the pot bottom to bulge downward due to uneven thermal expansion since the lower side of the pot bottom experiences higher temperature and hence expands more than the upper side. The reality, however, is just the opposite,
This central upward bulge is undesirable. For example, when frying an egg, one would like to place the egg near the center of the pot since the heat source from the stove gas flame or electric coil is normally close to the center of the pot. It is only natural to expect that the oil used for frying will stay near the center of the pot so that the oil can be heated more economically and the egg be fried more effectively. However, a central bulge in the pot will displace all the oil to the perimeter of the pot, leaving the central region hot and dry, smoking with smelly oil vapor, resulting in a scorched egg with broken yolk.
The reason why such bulges form is not difficult to understand. When the originally flat bottom of the pot is heated, it tends to expand. The cooler surrounding edge and vertical side of the pot, however, restrains it from free expansion, creating compressive stresses in the bottom plate. If the temperature gradient with respect to the plate thickness were uniform, the compressive stresses would be uniform, and no bulge would occur. However, during cooking, heat is extracted from the top surface of the plate, making the top surface temperature (say, of the order of the boiling point of water) substantially lower than the bottom surface temperature (which is closer to the stove flame temperature, several hundred degrees higher). Thus, the top surface tends to expand less than the bottom surface. Therefore, the top surface restraining compression .sigma..sub.1 is less than the bottom surface restraining compression .sigma..sub.2, forming a pressure gradient with respect to the plate thickness as shown in FIG. 7(a), which is the cross section of the free body of the flat part of the pot bottom. This non-uniform pressure field, linear or non-linear, can always be replaced by an equivalent force system of an axial force P and a bending moment M per unit length on the center plane of the plate along the circumferential boundary of the free body as shown in FIG. 7(b). It is this bending moment M, created by the larger compressive stress on the lower side of the plate relative to the upper that causes the plate to bulge upward, which, in turn, creates an eccentricity for the axial P to induce additional moment to aggravate the bulging. If, hypothetically, there exists such a material for making the pot that it does not creep under the high temperature of the stove flame, the bulge will disappear after the flame is removed and the pot is cooled off to the previous stress-free room temperature condition. The slight bulge during cooking would not be that intolerable. However, in actual life, the material does creep under stress at high temperature. Therefore, the metal normally used for making the pot tends to lose its ability to recover to its original flat condition, causing the so-called permanent set of the bulge. Although this permanent set is very small each time the pot is used, it is cumulative. After the pot is used hundreds of times, the bulge becomes prominent, rendering the pot less desirable, and often results in shortened pot life.
Since most material with good heat conducting properties for pot-making more or less creeps under the high flame temperature, it is almost impossible to completely avoid this bulge in the fiat bottom plate a pot. Fortunately, a slight bulge in the opposite direction would be a favorable condition for cooking. Take frying an egg as an example again. A slight depression at the center of the pot results in collecting the oil in the middle where the heat source is; and, when the fresh egg is placed there, its gravity presses the hot oil radially outward, forming a layer of hot oil beneath the egg, ideal for cooking. It is interesting to note that by introducing a suitable small central initial downward depression in the pot, the problem of upward bulge can be avoided. This initial bulge can be of a spherical of conical or any other single curvature surface.
FIG. 8(a) and FIG. 8(d) show the cross sections of a practically flat bottom pot with an exaggerated initial downward central depression or eccentricity e in the form of a spherical and conical cap respectively. First, according to the theory of structures, it has transformed a flat plate problem into a problem of shells; the latter is much stronger in resisting transverse deflections. Secondly, the initial eccentricity can induce an opposite bending moment to counterbalance the moment causing the upward bulge.
Just as in the case of a flat bottom pot, when heated from below, because of the thermal expansion and the non-uniform temperature gradient across the thickness of the bottom, there is the axial force P and the bending moment M acting at the perimeter of the free body of the spherical or conical bottom of the pot as shown in FIG. 8(b) and FIG. 8(e). Qualitatively, by taking a two-dimensional (instead of the actual three-dimensional shell problem which is more complicated) beam theory approach, at a point x from the perimeter as shown in FIG. 8(c) and FIG. 8(f), the bending moment there consists of two parts, namely, the temperature gradient induced M mentioned above and a second moment M'=Pe.sub.x created by the axial force P and the initial eccentricity e.sub.x at x. Note that the two moments M and M', are opposite in sense; and, if the initial eccentricity is sufficiently large, M' can be larger than M everywhere except near the perimeter of the freebody. Therefore, there will be no upward bulge, and most probably, there will be some additional but very limited downward deflection.
The minimum value of the initial downward eccentricity e is difficult to determine owing to the fact that it involves many variables including, but not limited to, the properties of the material used for the pot, the non-uniformity of the temperature of the stove flame, the flame temperature level and its size relative to that of the pot, the uneven heat extraction form the upper surface of the pot, the fixities of the side of the pot, etc. Fortunately, a precise determination is unwarranted since a larger e is harmless. For ordinary pots, a rough order-of-magnitude analysis (see next section) indicates that an initial central eccentricity of two times the thickness of the bottom plate of the pot is sufficient.
So, the problem of preventing the upward bulge of the flat bottom of an aged pot is resolved. Attention is now turned to the improvement on the second phenomenon of heat loss around the side of it. For the convenience of discussion, when the word "flame" is used, it shall mean the actual flame and the heated air around it from which heat can be transferred to the pot.
The reduction of heat loss can be accomplished in two aspects, namely, (a) by more efficiently transferring the heat from the flame to the pot, (b) by reducing the heat loss through radiation and convection from the flame into the surrounding air.
According to the basic principles of physics, the heat transferred to the pot from the flame is proportional to the area of contact of the flame with the pot, and also proportional to the time duration of the contact. Therefore, for efficient heat transfer, the flame-pot contact area should be made large and the contact duration should be made long.
For larger flame-pot contact area, the best way is to attach, to the underside and the side-wall surface of the pot, "fins" like those used in heat exchangers. Normally, such fins are made thin and dense for more efficient heat transfer. However, for pots and pans, they should be made more healthy in order to withstand the abusive conditions of cooking and washing.
For longer flame-pot contact duration, the fins should be attached to the bottom and side of the pot in a spiral pattern so as to channel the flame spiraling outward from the central part of the bottom to the outside edge and then helically upward along the side of the pot. Such channeling of the flame through the spiral/helical paths, without doubt, will increase the flame-pot contact duration.
It will also be appreciated that the central depression hereinabove suggested makes the pot unstable when placed on flat surfaces or stove tops. This can be rectified by having the fin depths tapered in the opposite direction to compensate for the depression to avoid this awkwardness.
To reduce the heat loss through radiation and convection around the pot, the most direct method is to attach a shield around it. For easy cleaning, such shield should be detachable. For free expansion of the pot and the fins, a slight gap should preferably be provided between the fins and the shield. The shield should be made of heat insulating material with or without metal lining. Exhaust vents should be provided in the returns of the upper edge of the shield or that of the pot where the shield meets the pot brim.
Thus, the improvement of heat loss reduction is accomplished, and an improved pot is created.
The longer pot life and the energy saved by using the pot of the present invention each time is insignificant. However, since a pot or pan is something almost everybody in the world uses, directly or indirectly, every day of the year, the cumulative saving is definitely significant. In the present energy and ecology conscious world, it is hoped that the pot of the present invention can make some contribution.
ORDER-OF-MAGNITUDE CALCULATION OF ECCENTRICITY
For a rough estimate of the required downward central eccentricity e needed to avoid the upward bulge of the circular flat bottom plate of a pot, the following assumptions are made for simplicity. (a) Since e is small, the bottom plate with the eccentricity e can still be treated like a plate in lieu of a shell. (b) No creep is considered. (c) The surface temperature of the bottom plate is the same throughout each of the upper or the lower surface, and the temperature gradient across the plate thickness is linear. Therefore, it becomes a circular plate under pure bending with uniform in-plane forces. (d) A small strip oft width b of the bottom plate through the center (a diameter strip) can be taken out and be treated with the small deflection beam theory (i.e. the Poisson effect is neglected or Poisson's ratio 1)=0). (e) The perimeter of the bottom plate or the Tends of the beam are first assumed to be unyielding, both axially and flexurally. Then they will be relaxed to their final situation; and these axial and flexural relaxations are assumed to be of similar proportions.
Notice that, while the other assumptions are reasonably close to reality, the last one may be off. In actuality, the plate boundary is neither completely rigid axially nor completely rigid flexurally. The degree of bending fixity at the plate perimeter depends on the flexural rigidity of the pot wall, while the degree of axial rigidity depends on the strain from hoop tension and the thermal expansion of the perimeter part of the bottom plate relative to the interior. It is very difficult to ascertain the exact amount of these relaxations. Since an upper limit, rather than the precise value, of e is desired here, to avoid complicated, involved, and probably not much mope accurate calculations, it will be assumed that the degrees of relaxation from the complete fixities with respect to the axial force and bending will be of the same proportion. The inaccuracy induced by this assumption will be compensated in later calculation by making e large so that the e-induced moment M' will become much larger than the bulging moment M.
Let t=plate thickness
E=Young's modulus PA1 .sigma.=Axial stress PA1 .beta.=Axial strain PA1 .alpha.=Coefficient of thermal expansion PA1 r=Radius of circular bottom plate of pot PA1 T.sub.t =Surface temperature of plate top above the stress-free room temperature PA1 T.sub.b =Surface temperature of plate bottom above the stress-free room temperature PA1 e.sub.x =Initial eccentricity at x from pot bottom the perimeter (See FIG. 8(b) and FIG. 8(e)); e.sub.x =e when x=r.
.sigma..sub.t =Stress at plate top surface PA2 .sigma..sub.b =Stress at plate bottom surface PA2 .beta..sub.t =Strain at plate top surface PA2 .beta..sub.b =Strain at plate bottom surface
FIG. 9(a) is the right half of FIG. 8(b) or FIG. 8(e); but, instead of a circular plate, it is considered to be a beam of depth t and width b with fixed ends. During cooking, assume the heat transfer has reached a steady state condition with the plate bottom temperature to be T.sub.b and that of the plate top T.sub.t. The stress at the beam end, shown in FIG. 9(a), can be considered as the sum of FIG. 9(b) and FIG. 9(c), which represent the axial force P and the bending moment M respectively. Then the axial force ##EQU1## and the positive moment caused by this axial force P and eccentricity e.sub.x at x is ##EQU2## which reaches its maximum value of ##EQU3## when x=r. And the negative moment caused by the uneven temperature in the plate is ##EQU4## At this point, the fixities of the beam ends ape relaxed. Since it is assumed that the axial force and bending moment will be relaxed by the same proportion, the relaxed force and moments become ##EQU5## where k&lt;1. Note that M.sub.1 is negative and constant, while M.sub.1 is positive and varies from zero at the boundary to its maximum value at center. When combined (see FIG. 10, the combined moment diagram), if M.sub.1 ' is greater than M.sub.1, the central portion of the plate will have net positive moment and hence will bulge downward. But there is always negative moment at the outskirt of the plate, which will cause the plate to bulge upward. In order to make this upward deflecting tendency insignificant, as well to compensate for the error induced by the assumption of proportional relaxation, M.sub.1 ' must be made much larger than M.sub.1.
Arbitrarily and conservatively let M.sub.1 '=12 M.sub.1. Then the negative moment is confined to the r/12 region at the perimeter of the plate (see FIG. 10). Therefore ##EQU6## If T.sub.b =800.degree. F. and T.sub.t =400.degree. F. above room temperature, then ##EQU7## If the extraction of best at the top of the plate is so complete (which is not likely in actual life) that it is kept at the stress-free room temperature, i.e., T.sub.t =0, then, EQU e=2t
Therefore, it can be concluded here that an e of the order of two times the plate thickness is sufficient to avoid the upward bulge of a pot or pan.